1. Law of Cosines
Section 5-6

2. Law of Cosines
Law of Cosines: Let ∆ABC be any triangle with sides a, b, c opposite angles A, B, C. Then:

3. Notes about Law of Cosines
The Law of Cosines is used for triangles which fit the SSS or SAS situations. Why must we use law of cosines here instead of the law of sines?
Once another angle is found it’s usually easier to use the law of sines to finish solving the triangle. However, realize that using the Law of Cosines is better to find missing angles, because the arccosine function will distinguish obtuse angles from acute angles. (Think back to the unit circle here!)

4. You try!
Ex 1: Solve ∆ABC given that a=11, b=5, and C=20°.

5. You try!
Ex 2: Solve ∆DEF given that d=9, e=7, and f=5.

6. Heron’s Formula
Let a, b, and c be the sides of ∆ABC, and let s denote the semiperimeter (a+b+c)/2. Then the area of ∆ABC is given by:

7. Heron’s Formula Ex:
Ex 4: Find the area of a triangle with sides 13, 15, and 18.

8. Triangle Area
Area of a Triangle: ∆Area =

9. Find the area
Ex 3: Find the area of a regular octagon inscribed inside a circle of radius 9 inches.